t-Structures on stable derivators and Grothendieck hearts

We prove that, given any strong and stable derivator and a t-structure in its base triangulated category D, the t structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one. We use this to show that the t structure being compactly generated implies that the coaisle is closed under directed homotopy colimits which, in turn, implies that the heart is an (Ab.5) Abelian category. If, moreover, D is a well-generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) t-structure has a generator. As a consequence, it follows that the heart of any compactly generated t-structure of a well generated algebraic or topological triangulated category is a Grothendieck Abelian category. & COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).